1. Kinetic Collision Detection for Convex Fat Objects
Mohammad Abam
Mark de Berg
Sheung-Hung Poon
Bettina Speckmann

2. Kinetic Data Structures
Simulate system of continuously moving objects.
Efficiently maintain discrete attributes of objects:
closest pair of objectst
convex hull
minimum spanning tree
binary space partition
...

3. Example: convex hull

4. Example: convex hull

5. Dynamic data structures
Allow insertions and deletions of objects at discrete times.
Not suitable for handling moving objects.

6. Time sampling approach
Choose fixed time step.
Update the positions of moving objects at each time step.
Update the data structure with the new positions of objects.

7. Time sampling approach
How to choose the proper time step?
Oversampling
Undersampling t t

8. Time sampling approach
Combinatorial changes occur in irregular patterns. t

9. Kinetic Data Structures
(Basch, Guibas, and Hershberger, 1998)
Use any static algorithm to compute the configuration function (i.e. the discrete attribute).
Turn the algorithm into a proof that this attribute is correct.
Animate this proof through time.

10. Example: convex hull Proof: a is to the left of bc
d is to the left of bc
b is to the right of ad
c is to the left of ad c d a b c b d a a c d b d a b c

11. Example: convex hull t1 Certificate Failure timed t1
Certificate Failure time
a is to the left of bc
d is to the left of bc
b is to the right of ad
c is to the left of ad t2 b
never
d is to the right of bc t1
b is to the left of ad t2
never

12. Kinetic Data Structures
Proof of correctness of attribute (certificates)
Priority queue (event queue)
Assumptions
A simple model for motion: each object follows a known flight plan with rational parameters.
Certificates are algebraic; failure is next largest root.

13. Kinetic Data Structures
Efficient KDS exist for
2d convex hull
Closest pair
Delaunay triangulation
Diameter, width of point sets
Collision detection between simple polygons in the plane, between two convex polyhedra in 3D, between multiple convex fat objects of similar sizes in 3D …

14. Kinetic Data Structures
Collision detection
Set of geometric tests (elementary certificates) that certify that the input objects are disjoint.

15. Balls rolling on a plane
Arbitrary sizes

16. Convex fat constant-complexity objects free-flying in 3D

17. Kinetic data structure properties
A KDS is called
compact if it uses little space in addition to the input.
responsive if the data structure invariants can be restored quickly after the failure of a certificate.
local if it can be updated easily if the flight plan for an object changes.
efficient if the worst-case number of events handled by the data structure is small compared to some worst case number of external events.

18. Locality

19. Related work
Same-size convex fat objects [Basch, Guibas, and Zhang, 1997]
Compactness O(n log2 n) certificates
Responsiveness each event can be handled in O(log3 n) time
Efficiency O(n2) events in the worst case
Two convex polyhedra [Erickson, Guibas, Stolfi, and Zhang, 1999] [Guibas, Xie, and Zhang, 2001]
We do arbitrary sizes!

20. Balls rolling on a plane
Balls Bi with center ci and radius ri.
Certificate of non-collision: dist(ci,cj) – ri – rj > 0 Bi Bj

21. Collision tree

22. Threshold disks
Smallest blue ball with radius = rmin

23. Threshold disks

24. Threshold disks

25. Threshold disks

26. Threshold disks
Threshold disk of radius Bi

27. Threshold disks

28. Threshold disks

29. Threshold disks

30. Threshold disks

31. Non-collision certificates
Lemma: Each point in the plane is contained in at most a constant number of threshold disks.
The number of non-collision certificates is O(n).

32. Kinetic maintenance

33. Kinetic maintenance
Replace threshold disks by threshold boxes.

34. Kinetic maintenance Replace threshold disks by threshold boxes.
Maintain two sorted lists on x- and y-coordinates of tangency points and threshold boxes.
O(n) certificates.

35. Kinetic maintenance
Lemma: Each point in the plane is contained in at most a constant number of threshold boxes.
The number of non-collision certificates is O(n).

36. Kinetic data structure properties
Balls rolling on a plane
Compactness O(n log n) certificates
Responsiveness each event can be handled in logarithmic time.
Efficiency O(n2) events in the worst case, assuming that the balls follow constant-degree algebraic trajectories.

37. Convex fat constant-complexity objects free-flying in 3D

38. Fat objects

39. Fat objects

40. Fat objects

41. Fat objects ri ro
Definition: An object is called ρ-fat if ro/ri < ρ.

42. Convex fat constant-complexity objects free-flying in 3D
Balls rolling on a plane:
threshold disks of radius
Balls free flying:
threshold disks of radius ri + rmin

43. Guarding fat objects

44. Guarding fat objects
Lemma For any ρ-fat object K, there is a set G(K) of O(ρ6) guarding points such that any ρ-fat object K’ that collides with K and is at least as large as K contains a point from G(K).

45. Guarding fat objects
Lemma For any ρ-fat object K, there is a set G(K) of O(ρ6) guarding points such that any ρ-fat object K’ that collides with K and is at least as large as K contains a point from G(K).

46. Problem: Hard to do efficiently
Guarding fat objects
Lemma For any ρ-fat object K, there is a set G(K) of O(ρ6) guarding points such that any ρ-fat object K’ that collides with K and is at least as large as K contains a point from G(K).
Approach
For each g ∈ G(K) maintain the object K(g) containing g
Solution Replace objects by their bounding boxes
Problem: Hard to do efficiently

47. maintain the smallest k objects whose bounding box contains g
Bounding boxes
A guard g can be contained in many bounding boxes but only the smaller objects can actually contain g
maintain the smallest k objects whose bounding box contains g

48. Making it work …
A guard g.

49. Making it work …
The unit cube centered at g.

50. Making it work …
A grid whose cells have edge length O(ρ2) grid cells.

51. Making it work …
Triangulate the grid.

52. Making it work …
Every triangle …

53. Making it work …
Every triangle defines an infinite cone.

54. Making it work … object K bb(K) = axis-aligned bounding box of K
guard g
Γ(g) = set of all cones for g
Lemma Let K(γ) be the set of all objects K whose centers lie in a cone γ and such that bb(K) contains the apex g of the cone. Suppose that one of these objects, K(g), contains g. Then K(g) must be among the smallest objects in K(γ).

55. Summary For each object K compute a set of O(ρ6) guards G(K).
For each guard g ∈ G(K) construct the set Γ(g) of O(ρ2) infinite cones with apex g.
For each cone γ∈Γ(g) maintain the smallest objects K*(γ) whose center lies inside γ and whose bounding box contains g.
Certify disjointness for K and all objects in the K*.

56. Maintenance
Need to maintain for each γ∈Γ(g) the collection K*(γ) of smallest objects whose centers are inside γ and whose bounding boxes contain g.
Box event
a bounding box starts or stops to contain a guard
Center event a center moves into or out of a cone
3 level range tree, three level segment tree, sorted lists, kinetic variants of range and segment trees

57. Kinetic data structure properties
Convex fat constant complexity objects free-flying in 3D
Compactness O(ρ2 n log6 n + ρ7n) certificates
Responsiveness each event can be handled in O(log6 n + ρ6) time.
Efficiency O(ρ13 n2) events in the worst case, assuming that the balls follow constant- degree algebraic trajectories.
linear number of candidate pairs

58. Kinetic data structure properties
Convex fat constant complexity objects free-flying in 3D
Compactness O(n log6 n) certificates
Responsiveness each event can be handled in O(log6 n) time.
Efficiency O(n2) events in the worst case, assuming that the balls follow constant- degree algebraic trajectories.
linear number of candidate pairs

59. Conclusions
First KDS for collision detection between multiple convex fat 3D objects of arbitrary sizes with a near-linear number of certificates.
Dependence on fatness parameter ρ is large.
Good bounds are possible in theory …
Is there a simple solution that achieves these bounds in practice?